Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/594

Rh ARITHMETIC Reduction. Compound a klition. Compound subtrac tion. Compound multipli cation. IV. In measuring cloth, the yard is divided into 4 quarters, the quarter into 4 nails of 2| inches. V. To measure land, the surveyor s chain of 100 links is used. The chain is 22 yards long, and 10 square chains make an acre. VII. The gallon and lower denominations are liquid measures ; the quart and those above it are for dry goods. VIII. The year, strictly speaking, is 365-24224 days. Every fourth year (leap year) has 366 days. 34. Reduction is the method of expressing quantities in a denomination lower or higher than that in which they are given. To reduce a higher denomination to a lower, multiply in succession by the numbers which show the times that the unit of each denomination (beginning with that given) contains the unit of the one next below it till the denomi nation is reached to which the quantity is to be reduced. If quantities in intermediate denominations are given, add each as its denomination is reached. To reduce a lower denomination to a higher, divide in succession by the numbers which show the times that the unit of each denomination (beginning with that given) is contained in the unit of the next above it, observing that the remainder after such division is of the denomination of the dividend. Sometimes the two processes are combined; thus, in reducing guineas to crowns, we multiply by 21, obtaining shillings, and then divide by 5. 35. Compound Addition and Compound Subtraction are the addition and subtraction of quantities expressed in more than one denomination. In Compound Addition, arrange the quantities according to their denominations, each under its proper heading ; add the lowest, and reduce the sum by division to the next higher, setting down the remainder, and carrying the quo tient ; add the others, including carriage, in the same way. Suppose that several sums of money are added, and the far things amount to 29, that is to 7|d., the |d. is set down and the 7d. carried to the pence column, and so in other cases. In Compound Subtraction, arrange the quantities as in Compound Addition, placing the greater amount over the other, and subtract, beginning with the lowest denomina tion. If in any case the lower number exceeds that above it, increase the latter by as many as make one of the next higher denomination, and afterwards add one to that denomination in the lower line. In subtracting, e.g., 1 qr. 25 Ib from 3 qr. 17 Bb, since 25 cannot be taken from 17, a quarter, i.e., 28 Bb, is added to the 17, making 45 ; from this 25 is subtracted, leaving 20 Bb, and the quarter &quot; borrowed &quot; is taken away again by being added to the 1 qr. The remainder is thus 1 qr. 20 Bb. Fractions of the lowest denomination are to be added or subtracted according to 18, 19; thus, 72, 8s. 45, 17s. 6|^d.=26, 10s. 8|d. 36. Compound Multiplication is multiplication in which the multiplicand is expressed in more than one denomination. When the multiplier does not exceed 12, multiply the different denominations by it, beginning with the lowest, and setting down and carrying as in Compound Addition. &quot;When the multiplier consists of several figures, multiply by each separately in the same way, taking them from right to left, and setting the result of each successive multiplication always one place further towards the left in each denomination, and add the results as thus arranged. In the accompanying example the above arrangement puts the product by 90 in the place of tens ; and the sums to be reduced to higher denomiua- | 63 12 254 10 572 13 &quot;~S~ = 4 times. =90 times. = 94 times. tions are 30 farthings, 104 pence, 148 shillings. If the multiplier be a composite number, we may multiply by the factors in succession. When the multiplier consists of two or more figures, the multiplication is often performed by the whole quantity at once. 37. Compound Division is division in which either the Com pom dividend or both dividend and divisor are expressed in division, more than one denomination. (1.) To find the amount that a given amount contains a given number of times, divide the highest denomination by the given number, reduce the remainder to the next lower denomination, adding the corresponding term of the dividend ; divide again, and proceed in the same way with the other denominations. The denominations of the quo tient correspond to those of the dividend. Let it be required, for instance, to divide 370, 16s. l|d. into 58 equal shares. As 58 shares of 6 each amount to 348, 6 is part of the quotient, and there remains 22, 16s. Ifd. Reducing 22, 16s. to shillings (456), we find that this gives 58 shares of 7s. each, with 50s. over. Similarly we obtain lOd. and 1 farthing, with 29 farthings over, which, since ff =, is just half a farthing for each of the 58 shares. The quotient then is 6, 7s. 10|d. |q. (2.) To find the number of times that one given amount contains another, reduce both to the same single denomina tion, and then divide the one by the other. To find, for example, how often 12s. 9|d. is contained in 171, 13s. 9d., since 12s. 9|d. = 615 farthings, and 171, 13s. 9d. = 164820 farthings, the number of times the second amount contains the first must be 164820 -f- 615, i.e., 268. It is to be observed that the quotient here is an abstract number. 38. In multiplying or dividing by fractions or mixed numbers, we follow the methods explained in 20, 21. As an 5 9 i T illustration, we give here the mul- &quot;^Ta&quot; tiplication of 24, 5s. 9|d. -fq. by 12) 121 9 f 62-j^-, where we may note that in JQ &quot;2 the division by 12 we have to 48 11 divide 2|- farthings, giving -^q. x 145 14 10 * T2- = -5T ( l-j an d tuat in adding the ,.,_,,. ~ z~j x fraction we have ^- + -A- = 2-^- + 39. Reduction of Fractions and Decimals. To find the Eednctic proper value of a fraction or a decimal of any denomina- of frac tion, multiply the fraction or the decimal by the numbers tions - in succession that reduce the denomination to lower de nominations. Thus, to find the value of |-- of a pound, it is manifest that this is |$ of 20 shillings, i.e., ff x 20s. = 12^s., or 12s. 3|d. ; and so in other cases. To find the value of a fraction of a quantity consisting of different denominations, we may either first reduce the quantity to one denomination, or we may multiply the compound quantity by the numerator, and divide by the denominator of the fraction; thus, of 5, 7s. ll|d. being the same as the ninth part of twice the amount, we may multiply by 2 and divide by 9, obtaining 1, 3s. llfd. -|q. as the result. 40. To reduce any amount to the fraction or the decimal of another denomination or amount, reduce both amounts to the same denomination, and write them as the terms of a fraction, the quantity of which the fraction is required being made the denominator. If the decimal is required, convert the vulgar fraction into a decimal. Thus, to reduce 13s. l|d. to the fraction and also to the decimal of a pound, since a pound contains 480 halfpence, a halfpenny is ^ of a pound, and therefore 13s. ld., i.e., 315 halfpence, is |i = = -65625. It is often sufficient to throw the expressions into the 4 &quot;2 T